1 Inference rules operatearbitrarily deep within formulae. ξ (A
−−
B )
2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.
Properties:
I Systems with cut are as efficient as their “shallow” counterparts.
I Exponential speedup for cut-free systems.
I There is a quasipolynomial cut-elimination procedure.
I Only has a restricted version of the subformula property.
Conjecture
All cuts can be eliminated in polynomial-time.
Deep inference
1 Inference rules operatearbitrarily deep within formulae. ξ (A
−−
B )
2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.
Properties:
I Systems with cut are as efficient as their “shallow” counterparts.
I Exponential speedup for cut-free systems.
I There is a quasipolynomial cut-elimination procedure.
I Only has a restricted version of the subformula property.
Conjecture
All cuts can be eliminated in polynomial-time.
Deep inference
1 Inference rules operatearbitrarily deep within formulae. ξ (A
−−
B )
2 The distinction between object and meta level is abolished. Proofs are top-down symmetric.
Properties:
I Systems with cut are as efficient as their “shallow” counterparts.
I Exponential speedup for cut-free systems.
I There is a quasipolynomial cut-elimination procedure.
I Only has a restricted version of the subformula property.
Deep inference
I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.
I While proof-search is now finitely branching, the “branching degree” is still unbounded, since we can now operate on any connective. Can we do better?
I QUESTION: How deep do we need to go?
Deep inference
I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.
I While proof-search is now finitely branching, the “branching degree”
is still unbounded, since we can now operate on any connective. Can we do better?
I QUESTION: How deep do we need to go?
Deep inference
I If conjecture is true then we have “finitary proof-search” with as short proofs as Frege/Gentzen with cut.
I While proof-search is now finitely branching, the “branching degree”
is still unbounded, since we can now operate on any connective. Can we do better?
I QUESTION: How deep do we need to go?
Surprise!
Symmetry alone is enough! For propositional logic, the same speedup in proof size can be attained just from top-down symmetry in the system.
ξ{D}
Surprise!
Symmetry alone is enough! For propositional logic, the same speedup in proof size can be attained just from top-down symmetry in the system.
ξ{D}
Technicalities
I What about the equations?
Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws?
Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules?
Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
Technicalities
I What about the equations? Turn them into inference rules and bound their depth too.
I Can we derive thedistributivity laws? Yes, in the presence of cocontraction.
I Can we still admit structural rules? Yes, and with no (significant) effect on proof complexity.
Theorem
Cut-free deep inference systems containing cocontraction can be restricted to allow just shallow inferences, with only polynomial increase in proof size.
So this means...
I Equates to augmenting a cut-free sequent system with the following two rules.
I No longer a sequent calculus; it breaks the asymmetry induced by the tree-like structure of sequent calculus proofs.
I Less restricted version of subformula property than that for regular deep inference.
I Worse for proof search than Gentzen systems, but much better than regular deep inference while still giving access to short proofs.
So this means...
I Equates to augmenting a cut-free sequent system with the following two rules.
I No longer a sequent calculus; it breaks the asymmetry induced by the tree-like structure of sequent calculus proofs.
I Less restricted version of subformula property than that for regular deep inference.
I Worse for proof search than Gentzen systems, but much better than regular deep inference while still giving access to short proofs.
Example
Thanks!